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Characterizing Realcompact Spaces As Limits of Approximate Polyhedral Systems

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Characterizing Realcompact Spaces As Limits of Approximate Polyhedral Systems Comment.Math.Univ.Carolin. 36,4 (1995)783–793 783 Characterizing realcompact spaces as limits of approximate polyhedral systems Vlasta Matijevic´ Abstract. Realcompact spaces can be characterized as limits of approximate inverse systems of Polish polyhedra. Keywords: approximate inverse system, approximate inverse limit, approximate resolu- tion mod P, realcompact space, Lindel¨of space, Polish space, non-measurable cardinal Classification: 54B25, 54C56, 54D30 1. Introduction In a series of recent papers various classes of spaces were characterized as spaces which admit resolutions (commutative or approximate), consisting of spe- cial classes of polyhedra. By a polyhedron we mean the carrier of a simpli- cial complex, endowed with the CW-topology. For pseudocompact spaces this was achieved in [6], for Lindel¨of and strongly paracompact spaces in [1], for n- dimensional spaces in [8] and [18] and for finitistic spaces in [12]. In this paper we obtain such characterizations for the class of realcompact spaces. In the liter- ature one can already find two characterizations of realcompact spaces in terms of inverse systems. The first one characterizes realcompact spaces as limits of inverse systems of Lindel¨of spaces ([15, Theorem 23]). The second one char- acterizes realcompact spaces as limits of inverse systems of Polish spaces ([3, Proposition 3.2.17]). Recall that a realcompact space can be defined as a space homeomorphic to a closed subspace of a product of copies of the real line R. Since R is topologically complete and topological completeness is preserved under direct products and closed subsets, every realcompact space is topologically complete. A separable space is called Polish if it is completely metrizable. A regular space is called Lindel¨of if its open coverings have countable refinements. Note that Polish spaces are Lindel¨of ([2, Theorem 3.8.1]) and Lindel¨of spaces are realcompact ([2, Theorem 3.11.12]). In both cases the inverse systems used to expand the space did not consist of polyhedra. Our aim is to expand realcompact spaces into polyhedral inverse systems. How- ever, these systems will be approximate inverse systems. The notions of an ap- proximate inverse system and its limit were recently introduced by S. Mardeˇsi´c, L. Rubin and T. Watanabe ([8], [11]). We state their basic definitions. 784 V. Matijevi´c Definition 1.1. An approximate (inverse) system is a collection X = (Xa, Ua,paa′ , A) consisting of: – a preordered set A = (A, <) which is directed and unbounded; – for each a ∈ A, a (topological) space Xa and a normal covering (mesh) Ua of Xa; ′ – for each comparable pair a<a in A, a (continuous) mapping paa′ : Xa′ → Xa, paa =1Xa is the identity mapping on Xa. These data must also satisfy the following three conditions: ′ ′′ (A1) (paa′ pa′a′′ ,paa′′ ) < Ua, whenever a<a <a ; ′ ′ (A2) (∀ a ∈ A) (∀ U ∈ Cov (Xa)) (∃ a >a) (∀ a2 >a1 >a ) (paa1 pa1a2 ,paa2 ) < U; ′ ′′ ′ − Cov ′′ 1 (A3) (∀ a ∈ A) (∀ U ∈ (Xa)) (∃ a >a) (∀ a >a ) Ua <paa′′ U. Here, for any two mappings f,g : X → Y and any covering V of Y , (f,g) < V means that, for every x ∈ X, there exists a V ∈V such that f(x),g(x) ∈ V . For coverings U, U′ of X, U′ < U means that U′ refines U. A normal (also called numerable) covering of a space X is any open covering of X which admits a subordinate partition of unity. The set of all normal coverings of X is denoted by Cov (X). Definition 1.2. An approximate map q from a space Y into an approximate system X , q : Y → X , is any collection q = {qa | a ∈ A} = (qa) of mappings qa : Y → Xa (called projections) such that: ′ (AS) For every a ∈ A and every U ∈ Cov (Xa) there exists an a >a such that ′′ ′ (qa,paa′′ qa′′ ) < U, whenever a >a . Definition 1.3. An approximate map p = (pa) : X → X is called a limit of X provided it has the following universal property: (UL) For any approximate map q : Y → X there exists a unique mapping q : Y → X satisfying pag = qa, for every a ∈ A. Since a limit space X is determined up to a unique homeomorphism, we often speak of the limit X of X and we write X = lim X . Let POL denote the collection of all polyhedra. Definition 1.4. An approximate resolution of a space X is an approximate map p : X → X = (Xa, Ua,paa′ , A) satisfying the following two conditions: (R1) (∀ P ∈ POL) (∀ V ∈ Cov (P )) (∀ f : X → P ) (∃ a ∈ A) (∀ a′ >a) (∃ g : Xa′ → P ) (gpa′ ,f) < V; (R2) (∀ P ∈ POL) (∀ V ∈ Cov (P )) (∃V′ ∈ Cov (P )) (∀ a ∈ A) ′ ′ ′ ′ ′′ ′ (∀ g,g : Xa → P ) (gpa,g pa) < V ⇒ (∃ a >a) (∀ a >a ) ′ (gpaa′′ ,g paa′′ ) < V. An approximate resolution of a space X can be characterized by conditions of a different kind. Instead of (R1) and (R2), which are often difficult to verify, more convenient are the following two equivalent conditions ([11, Theorem 2.8]). Characterizing realcompact spaces as limits of approximate polyhedral systems 785 ∗ −1 (B1) (∀ U ∈ Cov (X)) (∃ a ∈ A) (∃ V ∈ Cov (Xa)) pa V < U. ∗ ′ (B2) (∀ a ∈ A) (∀ U ∈ Cov (Xa)) (∃ a >a) paa′ (Xa′ ) ⊆ st (pa(X), U). Now we can state our main results. Theorem 1.5. For a paracompact space X the following statements are equiv- alent. (i) X is Lindel¨of. (ii) X admits an approximate resolution p : X → X = (Xa, Ua,paa′ , A), where all Xa are Polish polyhedra, all bonding maps paa′ are PL- mappings and all projections pa are surjections. (iii) X admits an approximate resolution p : X → X = (Xa, Ua,paa′ , A), where all Xa are Lindel¨of polyhedra. Theorem 1.6. For a topological space X the following statements are equivalent. (i) X is realcompact. (ii) X is the limit of an approximate inverse system X = (Xa, Ua,paa′ , A), where all Xa are Polish polyhedra and all bonding maps paa′ are PL. (iii) X is the limit of an approximate inverse system X = (Xa, Ua,paa′ , A), where all Xa are realcompact polyhedra. Since each paracompact space X is topologically complete, every polyhedral approximate resolution p : X → X = (Xa, Ua,paa′ , A) of X is a limit of X ([11, Theorem 3.1]). Therefore an approximate resolution in Theorem 1.5 is also a limit. 2. Polish, Lindel¨of and realcompact polyhedra For a complex K, let K0 denote the set of all vertices of K. Proposition 2.1. Let P = |K| be a polyhedron. Then the following statements are equivalent. (i) K0 is countable. (ii) P is separable. (iii) P is Lindel¨of. Proof: (i) ⇒ (ii). If K0 is countable, then |K| is a countable union of simplices. Since each simplex is a separable space we obtain (ii). (ii) ⇒ (iii) is obvious, since each separable paracompact space is Lindel¨of ([9, Corollary 3, Appendix 1] and [2, Corollary 5.1.26]). (iii) ⇒ (i). Suppose the contrary, i.e. that P = |K| is Lindel¨of, but K0 is uncountable. Then the open covering S = {st (v, K) : v ∈ K0} of P , formed by all stars of the vertices of K, has a countable refinement U. This implies the existence of a member U ∈U, which contains an uncountable subset of K0. Since U refines S and each member of S contains only one vertex of K, we obtain a contradiction. 786 V. Matijevi´c Proposition 2.2. A polyhedron P = |K| is Polish if and only if K is countable and locally finite. Proof: Let P = |K| be Polish. Proposition 2.1 implies that K is a countable set. On the other hand, metrizability of P implies that K is locally finite ([9, Example 1, Appendix 1]). Suppose now that K is countable and locally finite. Then P = |K| is separable and metrizable. The standard metric on |K| is defined by using the barycentric coordinates ([9, Appendix 1, § 1.3]). Since K does not contain any “infinite sim- plex”, this metric is complete ([5, Chapter 3, Lemma 11.5]), which shows that P is Polish. Note that each Polish polyhedron is itself a Polish ANR, i.e. a completely metrizable separable ANR. Therefore the family pPOL of all Polish polyhedra is contained in the family pANR of all Polish ANR’s. Remark 2.3. Let P = |K| be a separable polyhedron and let U ∈ Cov (P ) be an open covering of P . Let L be a subdivision of K such that the closed stars st (v, L) of the vertices v of L refine U, i.e. S = {st (v, L) : v ∈ L0} < U ([9, Theorem 4, Appendix 1]). Since L is a subdivision of K and card K0 ≤ ω, it follows that card L ≤ ω and therefore |L|m (the carrier of L with the standard metric topology) is a separable ANR ([5, Chapter 3, Theorem 11.3 and Lemma 11.4]). Let i : P → |L|m = Q be the identity mapping. It is well known that i has a homotopy inverse j : Q → P such that (ji,idP ) < S < U ([9, Theorem 10, Appendix 1]). This shows that the family sPOL of all separable polyhedra is approximately dominated by the family sANR of all separable ANR’s ([11, p. 599]). On the other hand, since each open covering of a Lindel¨of space has a countable star- finite refinement ([14, Chapter 5, 4B]), it can be proved that the family sANR is approximately dominated by the family pPOL. Therefore the families sPOL, sANR, pPOL and pANR are approximately equivalent. Proposition 2.4. A polyhedron P = |K| is realcompact if and only if τ = card(K0) is a non-measurable cardinal number.
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